### Advanced Probability Theory

Wednesday 11:15-12:50, Seminarraum 9 (OMP 1)

Thursday 11:15-12:50, Seminarraum 8 (OMP 1)

**Summary:** The lecture presents the most important
results and concepts of the modern probability theory, using
the measure-theoretic framework, in the context of infinite
sequences of random variables, i.e. stochastic processes in
discrete time. It gives an introduction to basic convergence
results for such sequences: laws of large numbers, central
limit theorem, large deviation for independent sequences;
convergence of stochastic series; martingale convergence
theorems; convergence of Markov chains. We discuss weak
convergence of probability measures on function spaces and
construct the Brownian motion by means of Donsker's
theorem.

**Literature:**

- R. Durrett: Probability theory & examples. (Duxbury Press, 1996)
- A. Klenke: Wahrscheinlichkeitstheorie. (Springer, 2008)
- D. Williams: Probability with martingales. (Cambridge University Press, 1991)
- P. Billingsley: Probability and measure. (Wiley, 1986)
- L. Breiman: Probability. (SIAM, 1992)
- A. Dembo: Probability Theory (lecture notes, Stanford University, download)

**Lecture notes:**

- Handwritten lecture notes for the same lecture given in SS2012 [PDF]
- Typed version in progress [PDF], comments and corrections wellcomed! (last updated on June 26)

**Exam questions:** [PDF]

### Proseminar to Advanced Probability Theory

Thursday 13:15-14:00, Seminarraum 8 (OMP1)

**Instructor:**Tobias Wassmer

**Summary:** The proseminar complements the lecture
with examples, exercises and short talks.

**Problems for solution:** TBD